I'm a Debian user. While I was trying apt-get install <package> I got this:

tar: ./es_DO/LC_ADDRESS: Cannot write: No space left on device
tar: ./es_DO/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./es_DO/LC_MONETARY: Cannot write: No space left on device
tar: ./es_DO/LC_TELEPHONE: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_ADDRESS: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_MESSAGES/SYS_LC_MESSAGES: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_MONETARY: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_TIME: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_COLLATE: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_TELEPHONE: Cannot write: No space left on device
tar: ./zh_CN.gbk/LC_NAME: Cannot write: No space left on device
tar: ./es_CO.utf8/LC_ADDRESS: Cannot write: No space left on device
tar: ./es_CO.utf8/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./es_CO.utf8/LC_MONETARY: Cannot write: No space left on device
tar: ./es_CO.utf8/LC_TELEPHONE: Cannot write: No space left on device
tar: ./my_MM/LC_ADDRESS: Cannot write: No space left on device
tar: ./my_MM/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./my_MM/LC_MESSAGES/SYS_LC_MESSAGES: Cannot write: No space left on device
tar: ./my_MM/LC_MONETARY: Cannot write: No space left on device
tar: ./my_MM/LC_CTYPE: Cannot write: No space left on device
tar: ./my_MM/LC_TIME: Cannot write: No space left on device
tar: ./my_MM/LC_TELEPHONE: Cannot write: No space left on device
tar: ./my_MM/LC_NAME: Cannot write: No space left on device
tar: ./br_FR@euro/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./br_FR@euro/LC_TIME: Cannot write: No space left on device
tar: ./nl_NL@euro/LC_ADDRESS: Cannot write: No space left on device
tar: ./nl_NL@euro/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./nl_NL@euro/LC_MONETARY: Cannot write: No space left on device
tar: ./nl_NL@euro/LC_TIME: Cannot write: No space left on device
tar: ./nl_NL@euro/LC_TELEPHONE: Cannot write: No space left on device
tar: ./aa_DJ.utf8/LC_ADDRESS: Cannot write: No space left on device
tar: ./aa_DJ.utf8/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./aa_DJ.utf8/LC_MESSAGES/SYS_LC_MESSAGES: Cannot write: No space left on device
tar: ./aa_DJ.utf8/LC_TIME: Cannot write: No space left on device
tar: ./zh_TW.utf8/LC_ADDRESS: Cannot write: No space left on device
tar: ./zh_TW.utf8/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./zh_TW.utf8/LC_TIME: Cannot write: No space left on device
tar: ./fur_IT/LC_IDENTIFICATION: Cannot write: No space left on device
tar: ./fur_IT/LC_TIME: Cannot write: No space left on device
tar: Exiting with failure status due to previous errors
dpkg: error processing locales-all (--configure):
subprocess installed post-installation script returned error exit status 2
configured to not write apport reports
Errors were encountered while processing:
locales-all
E: Sub-process /usr/bin/dpkg returned an error code (1)

So you are using apt-get as your package manager. And, apparently, apt-get uses tarballs as packages, so those need to be extracted using tar.

Thus, apt-get invokes tar, which proceeds with reading the archive and unpacking the files to the disk.

Unfortunately, when tar tries to unpack some of the files, it is told (by the operating system) that there is "No space left on device" (this usually means what it is supposed to mean — the filesystem tar is unpacking to is full). This happens to several files (tar does not give up until, it keeps processing the rest of the archive), and in the end it says it did not complete successfully, that there were errors (hence "Exiting with failure status due to previous errors").

The error is then handled by dpkg, which was called by apt-get, and was the tool which called tar, and each of these utils shows its own version of "something bad happened, look above to find why".

AFAIK, two reasons may cause the "No space left on device" error:
(1) The physical disk has been used up.
Run df -hk to check whether there is any free space left.
(2) There is enough disk space, but you have reached the maximum number that a file system can hold, because of huge numbers of files on the disk.
Run df -ih to see the used inode percentage, perhaps 100%?